Whether you are checking what day a birthday falls on, confirming the weekday of a historical event or planning around a date next year, this calculator names the day of the week for any date and writes out the arithmetic. You can see the work using either John Conway’s mental-math Doomsday rule or the formula known as Zeller’s congruence.
How to use the day of the week calculator
- Enter the date you want to look up. It defaults to today.
- Choose which method to show in Show the work using. Both always give the same answer.
- Read the weekday on the tape, along with the day of the year, the ISO week number, and whether it is the first, second, third, fourth or last such weekday in its month, which is handy for holidays defined that way.
Method 1: the Doomsday rule
Conway noticed that certain easy-to-remember dates always share a weekday within a year, which he called the doomsday. Find that weekday and you can count to any date.
Here c is the century (the first two digits of the year), y is the last two digits, and 0 = Sunday through 6 = Saturday. The century anchors repeat every 400 years: 1800s Friday, 1900s Wednesday, 2000s Tuesday, 2100s Sunday.
| Month | Doomsday | Memory aid |
|---|---|---|
| January | 3rd (4th in leap years) | 3 years out of 4, the 3rd |
| February | 28th (29th in leap years) | last day of February |
| March | 14th | Pi Day, 3/14 |
| April, June, August, October, December | 4/4, 6/6, 8/8, 10/10, 12/12 | even months repeat |
| May, September | 5/9 and 9/5 | “9 to 5 at the 7-Eleven” |
| July, November | 7/11 and 11/7 | the same line |
Method 2: Zeller’s congruence
Christian Zeller published this formula in the 1880s. It gives the weekday directly:
q is the day of the month, m the month (March = 3 through February = 14), K the year within the century and J the century. The result h counts from 0 = Saturday, 1 = Sunday up to 6 = Friday.
Worked examples
July 4, 1776 with the Doomsday rule. Century 17: (5 × 1 + 2) mod 7 = 0, a Sunday anchor. For y = 76: ⌊76/12⌋ = 6, remainder 4, ⌊4/4⌋ = 1. Doomsday = (0 + 6 + 4 + 1) mod 7 = 4, a Thursday. July 11 is a doomsday and July 4 is seven days earlier, so the Declaration of Independence was adopted on a Thursday.
January 1, 2000 with Zeller. January is month 13 of 1999, so K = 99 and J = 19. The terms are 1 + 36 + 99 + 24 + 4 + 95 = 259, and 259 mod 7 = 0, which is Saturday.
July 20, 1969 (the Moon landing). The 1900s anchor is Wednesday (3). For 69: 5 + 9 + 2 gives a doomsday of (3 + 16) mod 7 = 5, Friday. July 20 is nine days after July 11, and (5 + 9) mod 7 = 0, a Sunday.
Why weekdays drift from year to year
A common year has 365 days, which is 52 weeks plus 1 day, so a date usually moves one weekday later each year. After a February 29 it jumps two. That is why your birthday moves through the week in an uneven pattern; the birthday calculator lists the weekday of your next dozen birthdays. The full Gregorian cycle is 400 years, or exactly 146,097 days, which divides evenly into 20,871 weeks. That is why every calendar repeats with the same weekdays every 400 years.
If you need the ISO week number of a date as well, see the week number calculator, and for leap-year rules see the leap year calculator.
Frequently asked questions
What day of the week was July 4, 1776?
A Thursday. With the Doomsday rule, the 1700s anchor is Sunday and the 1776 doomsday works out to Thursday. July 11 is a doomsday, and July 4 is exactly one week earlier, so it is also a Thursday.
Why do January and February count as months 13 and 14 in Zeller's formula?
Zeller's congruence treats the year as starting in March so that the leap day, February 29, falls at the very end of the year. That keeps the month term regular. January and February are counted as months 13 and 14 of the previous year.
What are the doomsdays for each month?
In any year these dates share a weekday: January 3 (January 4 in leap years), February 28 (29), March 14, April 4, May 9, June 6, July 11, August 8, September 5, October 10, November 7 and December 12.
Does it work for dates before 1582?
It gives the weekday on the proleptic Gregorian calendar, which extends today's rules backward. Historical documents from before the switch used the Julian calendar, whose dates differ by 10 days or more, so convert the date first if you are checking an old record.